Ill-posedness for the gCH-mCH equation in Besov spaces

Abstract

In this paper, we consider the Cauchy problem to the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/Modified Camassa–Holm equation and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation. We prove the ill-posedness for the Cauchy problem of the gFQXL/gCH-mCH equation in \(B^s_{p,\infty }\) with \(s>\max \{2+1/p, 5/2\}\) and \(1\le p\le \infty \) in the sense that the solution map to this equation starting from \(u_0\) is discontinuous at \(t = 0\) in the metric of \(B^s_{p,\infty }\).